Update consecutive-numbers-sum.py

Author: kamyu104

Python/consecutive-numbers-sum.py

@@ -10,8 +10,16 @@ def consecutiveNumbersSum(self, N):
         # x + x+1 + x+2 + ... + x+l-1 = N = 2^k * M, where k >= 0 and M is odd
         # => l*x + (l-1)*l/2 = 2^k * M
         # => x = (2^k * M - (l-1)*l/2) / l = 2^k * M/l - (l-1)/2 is a positive integer
-        # => l could be 2^(k+1) or any odd factor of M (only one of 2^(k+1) and M is possible)
+        # => l could be 2^(k+1) or any odd factor of M (only one of 2^(k+1) and M must be possible, proof as follows)
         #    s.t. x = 2^k * M/l - (l-1)/2 is a positive integer, and also unique
+        #
+        #     proof:
+        #     if l = 2^(k+1) is impossible
+        #     <=> 2^k * M/(2^(k+1)) - (2^(k+1)-1)/2 <= 0
+        #     <=> M - 2^(k+1) <= -1
+        #     <=> 2^(k+1) >= M+1
+        #     <=> 2^k * M/M - (M-1)/2 >= 1, l = M is possible
+        #    
         # => the answer is the number of all odd factors of M
         # if prime factorization of N is 2^k * p1^a * p2^b * ..
         # => answer is the number of all odd factors = (a+1) * (b+1) * ...